Strongly flat modules via universal localization

Abstract

In this paper, we investigate a non-commutative version of strongly flat modules, which is based on the concept of universal localization introduced by Cohn. We consider a set $\ddot{I}ƒ$ consisting of maps of finitely generated projective $R$-modules, where $R$ is not necessarily a commutative ring. Let $R_{\ddot{I}}ƒ$ denote the universal localization of $R$ with respect to $\ddot{I}ƒ$. The class of $\ddot{I}ƒ$-strongly flat modules is defined as the left class in the cotorsion pair generated by $R_{\ddot{I}}ƒ$. We examine the homotopy category of $\ddot{I}ƒ$-strongly flat modules and demonstrate that the thick subcategory ${\mathcal{S}}_{\ddot{I}}ƒ$, consisting of acyclic complexes, wherein all syzygies are $\ddot{I}ƒ$-strongly flat, forms a precovering class within this homotopy category. This implies that the quotient map from $\mathbb{K}({σ\mbox{-}\mathcal{SF}})$ to $\mathbb{K}({σ\mbox{-}\mathcal{SF}})/\mathscr{S}_σ$ always has a fully faithful right adjoint.

Comments are welcome